Mark, that is a nice summary. If the Black Sky rocket had a short burn-time, then there would be no appreciable increase in performance.
From 1-D compressible gas dynamics it is well known that above the critical pressure ratio there will be sonic flow at the throat. Roughly, the pressure at the throat is about half the chamber pressure (it depends upon the propellant, specific heat ratio, etc.). If the nozzle has only a throat (for example the old model Jetex motors with just a hole in the aft end of the case), all the opportunity for using the high pressure exhaust gas to produce thrust is lost. Usually, the increase in thrust for the divergent part of the nozzle is defined by the thrust coefficient for which there is a theoretical formula derived from 1-D compressible gas dynamics. So, both the bell nozzle and the aerospike have a throat, but how the thrust is extracted from the geometry of the nozzle is different in the supersonic expansion of the gas flow. I tend to think that a linear spike might do better than the cylindrical (or annular) spike, because there is more nozzle surface area for a given amount flow to extract thrust from the high pressure gas. I knew someone about 10 or 15 years ago that made some CFD runs for the cylindrical spike geometry and he told me that he was not seeing the performance increases that everyone was expecting.
